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Theorem dihval 35335
Description: Value of isomorphism H for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 3-Feb-2014.)
Hypotheses
Ref Expression
dihval.b 𝐵 = (Base‘𝐾)
dihval.l = (le‘𝐾)
dihval.j = (join‘𝐾)
dihval.m = (meet‘𝐾)
dihval.a 𝐴 = (Atoms‘𝐾)
dihval.h 𝐻 = (LHyp‘𝐾)
dihval.i 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihval.d 𝐷 = ((DIsoB‘𝐾)‘𝑊)
dihval.c 𝐶 = ((DIsoC‘𝐾)‘𝑊)
dihval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihval.s 𝑆 = (LSubSp‘𝑈)
dihval.p = (LSSum‘𝑈)
Assertion
Ref Expression
dihval (((𝐾𝑉𝑊𝐻) ∧ 𝑋𝐵) → (𝐼𝑋) = if(𝑋 𝑊, (𝐷𝑋), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))))
Distinct variable groups:   𝐴,𝑞   𝑢,𝑞,𝐾   𝑢,𝑆   𝑊,𝑞,𝑢   𝑋,𝑞,𝑢
Allowed substitution hints:   𝐴(𝑢)   𝐵(𝑢,𝑞)   𝐶(𝑢,𝑞)   𝐷(𝑢,𝑞)   (𝑢,𝑞)   𝑆(𝑞)   𝑈(𝑢,𝑞)   𝐻(𝑢,𝑞)   𝐼(𝑢,𝑞)   (𝑢,𝑞)   (𝑢,𝑞)   (𝑢,𝑞)   𝑉(𝑢,𝑞)

Proof of Theorem dihval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dihval.b . . . 4 𝐵 = (Base‘𝐾)
2 dihval.l . . . 4 = (le‘𝐾)
3 dihval.j . . . 4 = (join‘𝐾)
4 dihval.m . . . 4 = (meet‘𝐾)
5 dihval.a . . . 4 𝐴 = (Atoms‘𝐾)
6 dihval.h . . . 4 𝐻 = (LHyp‘𝐾)
7 dihval.i . . . 4 𝐼 = ((DIsoH‘𝐾)‘𝑊)
8 dihval.d . . . 4 𝐷 = ((DIsoB‘𝐾)‘𝑊)
9 dihval.c . . . 4 𝐶 = ((DIsoC‘𝐾)‘𝑊)
10 dihval.u . . . 4 𝑈 = ((DVecH‘𝐾)‘𝑊)
11 dihval.s . . . 4 𝑆 = (LSubSp‘𝑈)
12 dihval.p . . . 4 = (LSSum‘𝑈)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dihfval 35334 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))))
1413fveq1d 6090 . 2 ((𝐾𝑉𝑊𝐻) → (𝐼𝑋) = ((𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))))‘𝑋))
15 breq1 4580 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
16 fveq2 6088 . . . 4 (𝑥 = 𝑋 → (𝐷𝑥) = (𝐷𝑋))
17 oveq1 6534 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥 𝑊) = (𝑋 𝑊))
1817oveq2d 6543 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑞 (𝑥 𝑊)) = (𝑞 (𝑋 𝑊)))
19 id 22 . . . . . . . . 9 (𝑥 = 𝑋𝑥 = 𝑋)
2018, 19eqeq12d 2624 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑞 (𝑥 𝑊)) = 𝑥 ↔ (𝑞 (𝑋 𝑊)) = 𝑋))
2120anbi2d 735 . . . . . . 7 (𝑥 = 𝑋 → ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) ↔ (¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋)))
2217fveq2d 6092 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐷‘(𝑥 𝑊)) = (𝐷‘(𝑋 𝑊)))
2322oveq2d 6543 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐶𝑞) (𝐷‘(𝑥 𝑊))) = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))
2423eqeq2d 2619 . . . . . . 7 (𝑥 = 𝑋 → (𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))) ↔ 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))
2521, 24imbi12d 332 . . . . . 6 (𝑥 = 𝑋 → (((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))) ↔ ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))))
2625ralbidv 2968 . . . . 5 (𝑥 = 𝑋 → (∀𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))) ↔ ∀𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))))
2726riotabidv 6491 . . . 4 (𝑥 = 𝑋 → (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))) = (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))))
2815, 16, 27ifbieq12d 4062 . . 3 (𝑥 = 𝑋 → if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))) = if(𝑋 𝑊, (𝐷𝑋), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))))
29 eqid 2609 . . 3 (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))) = (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))))
30 fvex 6098 . . . 4 (𝐷𝑋) ∈ V
31 riotaex 6493 . . . 4 (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))) ∈ V
3230, 31ifex 4105 . . 3 if(𝑋 𝑊, (𝐷𝑋), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))) ∈ V
3328, 29, 32fvmpt 6176 . 2 (𝑋𝐵 → ((𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))))‘𝑋) = if(𝑋 𝑊, (𝐷𝑋), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))))
3414, 33sylan9eq 2663 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑋𝐵) → (𝐼𝑋) = if(𝑋 𝑊, (𝐷𝑋), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  ifcif 4035   class class class wbr 4577  cmpt 4637  cfv 5790  crio 6488  (class class class)co 6527  Basecbs 15641  lecple 15721  joincjn 16713  meetcmee 16714  LSSumclsm 17818  LSubSpclss 18699  Atomscatm 33364  LHypclh 34084  DVecHcdvh 35181  DIsoBcdib 35241  DIsoCcdic 35275  DIsoHcdih 35331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-dih 35332
This theorem is referenced by:  dihvalc  35336  dihvalb  35340
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